Building traversable wormholes from Casimir energy and non-local couplings

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UNIVERSITY OF AMSTERDAM

Master Thesis

Building traversable wormholes from

Casimir energy and non-local

couplings

by

Theodora Nikolakopoulou

11410310

Main Supervisor:

dr. Ben Freivogel

Second Supervisor:

dr. Diego Hofman

July 16, 2018

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Abstract

The main purpose of this project is finding ways to construct traversable wormholes (TW) and studying various aspects of them. It is now thirty years since Thorne and Morris [1] understood that wormholes require the presence of exotic matter that violates the null energy condition. In order to produce this necessary negative energy density we mainly follow two different approaches.

First, we explore ways of making a wormhole traversable by using Casimir energy [2]. We study the work of Butcher [3], in which he constructs a long throat TW by using a non-minimally coupled quantum scalar field, and we also make an attempt to construct an asymptotically AdS wormhole using a photon field.

The bigger part of this work is dedicated to another approach one can follow in order to make a TW. Recently, Gao, Jafferis and Wall [4] showed that by coupling two asymptotic boundaries of a maximally extended BTZ black hole, the Einstein-Rosen bridge connecting the two asymptotic regions can be rendered traversable. In this thesis, we study extensively these non-local couplings, first in flat space, and then in the case of the BTZ black hole. We then perform explicit calculations in order to make sure that any signal we send through this wormhole will indeed reach the other side safely and that no violent events, such as the creation of another black hole, will take place. Furthermore, we make some estimates about how much information we can send through, from the bulk point of view, as well as from the quantum teleportation point of view.

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1 Introduction

1.1 Wormhole origins and 1

st

Renaissance

One of the most popular and exciting science-fiction concepts is that of a wormhole. Numer-ous movies, series and books include such spacetime shortcuts that allow travellers to cover distances that otherwise would take many lifetimes to travel. But the burning question is: can we actually do this? Does physics allow for the existence of such objects?

Physicists have been puzzling over this question for almost a century. The simplest theoretical wormholes are non-traversable, which means that even though they can exist we cannot send anything through. The main problem is that they connect regions of space that are spacelike separated. But, from a practical point of view there are more obstacles, such as horizons and curvature singularities. If an astronaut decided to take a trip down a black hole, all we would ever see is her moving slower and slower but never reaching the horizon. Moreover, even if the astronaut could somehow escape a curvature singularity, the tidal effects close to it would be extreme enough to tear her apart. All the above, as well as other problems1 _{of the first theoretical wormholes, were discouraging physicists from taking}

them seriously.

However, in 1988, Morris and Thorne found a way to construct a traversable wormhole with pleasing characteristics. In order to avoid having both horizons and naked singularities, which is the least one can ask in order to have a well-behaved object, they chose to con-sider wormholes that have no curvature singularity. However, they did not follow the usual approach (picking a Lagrangian with fields that hopefully support a wormhole, finding the stress tensor and solving the Einstein equations), which was proven not to be a fruitful path. They did the process in reverse: they chose a suitable metric describing a well-behaved wormhole, they found the Einstein tensor and deduced what the stress tensor should be. What they found was that the matter near the wormhole throat should not be the ordinary matter that we constantly stumble upon in our universe, but an “exotic” kind of matter that violates the null energy condition, along with all the other energy conditions (which we will further explain in 2.1, 2.2). Now, if we were trying to make a purely classical wormhole (a wormhole supported only by classical fields) the conclusions of Morris and Thorne should make us sigh in despair. However, we know that in quantum field theories (QFTs) the energy conditions we previously mentioned are not true any more, we can measure negative energy locally, so not all is lost.

1_{One of the other problems we refer to is, for example, naked singularities, which were considered in order}

to avoid having horizons in the way. However, such things violate the cosmic censorship hypothesis, which leads to the failure of determinism.

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1.2

2

nd

Renaissance

Many physicists throughout the years have tried to build traversable wormholes using dif-ferent fields and techniques, but the construction that stood out and led to the comeback of traversable wormholes was that of Gao, Jafferis and Wall (GJW) [4], in 2016. In this work, they constructed a traversable wormhole in Anti-de-Sitter (AdS) spacetime. Their set-up was the maximally extended AdS-Schwarzschild black hole2 _{in three dimensions (otherwise}

called the BTZ black hole). This geometry has two asymptotically AdS regions, that are connected by a non-traversable wormhole which collapses into a singularity. Starting from this geometry, they coupled the right and left boundaries of the black hole at some time t0,

by adding in the action a term of this form: δS =

Z

dt dx hOR(t, x)OL(−t, x), (1.1)

where O is an operator dual to a scalar field ϕ in the bulk, and h is the coupling constant. This resulted in the propagation of negative energy shock waves in the bulk. Thus, the quantum matter stress tensor violates the averaged null energy condition. So, if we had sent a light-ray really early (almost hugging the horizon) from the left/right boundary towards the right/left boundary, it would not end up in the singularity. Instead, it would in principle pass through the horizon, gain a time advance due to the encounter with the negative energy density, and reappear at the right/left boundary. Thus, the wormhole is rendered traversable. Of course, in real life it is not possible to connect two asymptotic regions since they are spacelike separated. However, in a lab we could imagine building two copies of a CFT on two plates and connect them with some “wire”. Thus, to consider a theory with such a non-local coupling term is certainly something sensible.

This publication was followed by another, authored by Maldacena, Stanford and Yang (MSY)[8], where they explored a similar set-up in AdS2. It is worth mentioning that in their

paper, they constructed a quantum teleportation protocol picture. Quantum teleportation is the process during which we transmit a quantum state over long distances, while having to transport only classical information. This process also requires previously shared entan-glement between the sending and receiving region. So, we are actually moving one qubit from one place to the other without having to physically transport the underlying particle to which that qubit is normally attached. The actual protocol goes as follows. First, we split an EPR pair and give one qubit to Alice and the other one to Bob. Alice also has a qubit that she wants to teleport (we’ll call it the teleportee). She makes a Bell measurement of the EPR pair qubit and the qubit to be teleported, and obtains a result. Then, she sends this result to Bob through the classical channel. Finally, Bob modifies his qubit in order to make it identical to the teleportee. MSY made a variation in the set-up of GJW, in order to

2_{Of course, one of the reasons to consider a black hole in AdS, instead of any other spacetime, is that}

we can think of it in the context of AdS/CFT, in which the BTZ is dual to two copies of a CFT in the thermofield double state

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make it the same as the procedure we just described. Using this picture, they also calculated some bounds on the information that can be sent through the wormhole, which is going to be of interest for the purposes of this thesis. More publications inspired by the idea of GJW followed [9],[10], [11], [12], [13] and thus, we might say that they sparked a second Renaissance for traversable wormholes.

What is more, the set-up of GJW is not interesting just because of the obvious result of making an Einstein-Rosen bridge traversable. It is important because it provides a model of how a signal, and information in general, can escape a black hole.

1.3 Summary of Results

The main focus of this thesis is to study how to make traversable wormholes using non-local couplings. In order to understand how they work we first apply them in the simplest case we could think of: the free massless scalar in 1+1 dimensional flat spacetime. We add to the action the interaction term δS = −gφLφR, where φL,R is the field operator

evaluated at uL,R, vL,Rrespectively, and calculate the stress tensor. What we find is that the

expectation value of the energy density, to first order in the coupling constant g, had the form of two positive and two negative energy shock waves propagating in spacetime, as in the left sub-figure below. We then smear out our sources in a diamond-shaped area around (uL,R, vL,R) and re-calculate the expectation value of the energy density to first order in g.

This time, instead of localized shock waves, we obtain extended strips of positive/negative energy density. (uL, vL) (0, 0) (uR, vR) a b c d e f g _h _i (0, 0) (uL, vL) (uR, vR)

Figure 1: On the left we see the configuration without the smearing, whereas on the right we see the configuration when we smear our sources. The blue/red lines or strips always represent the negative/positive shock waves, respectively.

It’s easy to see that in both cases (non-smeared and smeared) the integral of the stress tensor, along constant v (with vL < v < vR) or u (with uL < u < uR), is negative and thus

we violate the average null energy condition (ANEC). The motivation to smear our sources is that we want to calculate the second order correction to the expectation value of the stress tensor. In this calculation, we encounter the IR divergences of the scalar field, which are cured if we smear our sources. For example, our result for the second order term in strip f is: hTuuif = − g 32A2_π log uL− u − A uL− u + A − g 2

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where A is half the side of the diamond, and µ is the IR-cutoff. Furthermore, we check whether the Quantum Inequalities (the other popular way of restricting negative energies) are true for such a set-up. We find that (1.2) is violating these as well.

Of course, the most interesting part is when we apply this to the case of the BTZ black hole. In our case, instead of smearing our operators OL and OR like GJW, we insert them

on a single instant of time. The reason behind this choice is that we want to have an analytic expression for the stress tensor, which we calculate to be:

hTU Ui = − ∆ sin π∆ 22∆+1/2_π3/2 Γ(1 − ∆) Γ(3₂− ∆) h ` U−(∆+1)√2 (1/2 − ∆) (1 − U/U0)∆+1/2(1 + U/U0)1/2 U2 0 + 1 U0 −(∆+1) F1 −∆; 1 2, ∆ + 1; 1 2 − ∆; U − U0 U + U0 , U − U0 U 1 + U2 0 ! , (1.3)

where ∆ is the scaling dimension of O and U0 is the point of insertion at the boundary of

AdS. We also find that the integral of the stress tensor for ∆ < 1/2 is negative (if we choose h > 0) and thus, hTU Ui violates the ANEC. Hence, the Einstein-Rosen bridge is rendered

traversable. In addition, we calculate how much the wormhole opens-up, or otherwise the shift that a signal would take upon collision with this negative energy, and find it to be of order Planck scale.

In order for a signal to pass through such a small opening it has be highly boosted. However, if this signal is very energetic we have to make sure that upon collision with the negative energy shock wave there are no stringy effects that we should take into consideration, and that no new black holes are created. For this reason, we assume that the signal and the negative energy shock wave are particles, and we calculate the center of mass energy of the collision. We find it to be of order 1√`P. Thus, we believe that we should not worry

about the aforementioned possible complications and moreover, there is even room to send more than just one particle through. We calculate the number of different and same species particles that we can send through until we reach Planck energy, and we find them to be ndif f max = h ` `P and n same max = q h_``

P respectively, which are both very big numbers. So, from the

bulk point of view it seems that we are allowed to send a lot of particles. In other words, if we qubits to these particles we can send a big amount of information to the other side. This seems to be clashing with the result of MSY. As we will see, if we naively use their result in our case, we conclude that we are only allowed to send less than one particle.

1.4 Outline

The outline of this thesis is as follows. In chapter 2, we provide some background knowledge needed for the better understanding of this thesis, such as the energy conditions, the Casimir effect and a brief introduction in AdS/CFT. Next, in chapter 3 we review the work of Butcher [3], in which he constructs a traversable wormhole that is supported by its own Casimir

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energy, using a massive, non-minimally coupled, quantum, scalar field. Moreover, we make an attempt to construct an asymptotically AdS traversable wormhole using a photon field.

In chapter 4 we switch gears and focus on the BTZ black hole, which is the main set-up that we are going to work with in the rest of the thesis. We present some of its basic characteristics, its propagators and its CFT dual. In addition, in chapter 5, we describe the derivation of shock waves in the BTZ geometry, which is an essential concept for what follows. Next, in chapter 6, we present an easy way of acquiring negative energy densities in QFT, by the use of non-local couplings, which is essential in order to violate ANEC. When this idea is applied in the case of the BTZ black hole even more interesting things happen. So, in chapter 7 we how to make a wormhole traversable using this idea. We follow the chronological order and we first review the set-up of GJW and then that of MSY. Finally, we choose a slightly different interaction term than the one of GJW, we calculate the matter stress tensor at the horizon and we find that it violates the ANEC. We then do some checks in order to make sure that the signal will travel through the wormhole safely and we calculate the maximum number of particles that we can send through.

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2 Preliminaries

2.1 Energy conditions

One concept that we have to get acquainted with in order to begin understanding wormholes is the energy conditions. The essence of GR can be captured by the Einstein equations:

Gµν = 8πGNTµν, (2.1)

where Gµν is the Einstein tensor and Tµν is the stress-energy tensor. The l.h.s. represents

the curvature of spacetime and is determined by the metric and the r.h.s. represents the matter/energy content of spacetime. So, the Einstein equations can be summarized as the main relation between matter and the geometry of spacetime.

Despite their elegance, the Einstein equations have a great deal of arbitrariness when it comes to deciding what Tµν is going to be. Since all metrics satisfy Einstein equations,

we can choose any metric we like, calculate the Einstein tensor and then demand that Tµν

is proportional to Gµν. However, this does not necessarily mean that the stress tensor we

found is going to describe a realistic source of energy.

In order to make sure that the stress tensors we deal with are physical we have to impose some restrictions, namely, the energy conditions. As Carroll [5] explains: “The energy conditions are coordinate-invariant restrictions on the energy-momentum tensor ”. In order to have coordinate-invariant quantities we construct scalars that contain the stress tensor by contracting it to timelike or null vectors. There are many different energy conditions that apply to different circumstances. In order to gain more intuition we will use the stress tensor of the perfect fluid, which is:

Tµ = (ρ + p)UµUν + pgµν, (2.2)

where Uµ is the four-velocity of the fluid, ρ is the energy density and p the pressure. So, let’s

now see the most frequently used energy conditions:

1. The Null Energy Condition (NEC): Tµν`µ`ν ≥ 0 for all null vectors `µ. This condition

is the hardest one to violate. For the perfect fluid it implies that ρ + p ≥ 0.

2. The Weak Energy Condition (WEC): Tµνtµtν ≥ 0 for all timelike vectors tµ. WEC

includes NEC and it for the perfect fluid it implies that ρ ≥ 0 and ρ + p ≥ 0.

3. The Dominant Energy Condition (DEC): Tµνtµtν ≥ 0 (WEC) and TµνTνλtµtλ ≤ 0,

for all timelike vectors tµ_{. The second part of the condition, namely T}

µνTνλtµtλ ≤ 0,

means that Tµνtµ is a non-spacelike vector. As we see DEC includes WEC. For the

perfect fluid DEC means that ρ ≥ |p|, i.e. the energy density is bigger or equal than the absolute value of the magnitude of pressure.

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4. The Null Dominant Energy Condition (NDEC): Tµν`µ`ν ≥ 0 (WEC) and TµνTνλ`µ`λ ≤

0, for all null vectors `µ_{. The second part of the condition, namely T}

µνTνλ`µ`λ ≤ 0,

means that Tµν`µ is a non-spacelike vector. The NDEC is the DEC for null vectors

only. The densities and pressures allowed are the same as for the DEC, except negative energy densities are allowed as long as p = −ρ.

5. The Strong energy condition (SEC): Tµνtµtν ≥ 1₂Tλλtσtσ, for all timelike vectors tµ.

SEC does not imply WEC. However, it implies NEC but at the same time it does not allow for very large negative pressures. For the case of the perfect fluid SEC means ρ + p ≥ 0 and ρ + 3p ≥ 0.

2.2 Average Energy Conditions

The energy conditions are in general true for classical matter. There are some exceptions, for example, it is possible to violate the SEC in the case of a classical free scalar field, but especially the WEC and the NEC are indeed always obeyed.

However, upon entering the quantum realm these conditions, as well as the rest of the energy conditions, cease to be true and observers can measure negative energy density. Examples where this happens is the Casimir effect [2] and the squeezed photon states [14], both of which have been experimentally observed. Moreover, the existence of negative energy density is required for the Hawking evaporation of black holes [15]. However, having no restrictions on how much negative energy density we are allowed to observe, can result to the violation of cosmic censorship [16],[17] and the second law of thermodynamics [18], [19]. As a result, over the recent years there have been significant efforts in finding reasonable constraints.

There have been two main approaches. The first one is the quantum inequalities, first introduced by Ford [20] which are constraints on the magnitude and duration of the negative energy fluxes and densities, measured by an inertial observer. The second one, which is the subject of this chapter, is the average versions of the energy conditions, first discussed by Tipler [21]. He thought of integrating the WEC over a whole worldline of some observer. This can be done for other energy conditions as well. The success of this method is that these non-local conditions do hold for quantum field theories, unlike their local counterparts. Two of the most popular averaged energy conditions are the following:

1. The Averaged Null Energy Condition (ANEC): R_γTµν_`µ_`ν_{dλ ≥ 0, for all null vectors}

`µ_{. We integrate over a null curve γ. Moreover, λ is a generalized affine parameter for}

the null curve.

2. The Averaged Weak Energy Condition (AWEC):R_γTµν_tµ_tν_{dτ ≥ 0, for all timelike}

vec-tors tµ_{. We integrate over a timelike curve γ and τ is the proper time parametrization}

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2.3 Proofs of ANEC

All these energy conditions that we previously mentioned have been motivated by GR, and in this context they are considered to hold in any spacetime, curved or flat. However, it is highly non-trivial to prove them for quantum field theories, even in the case of flat spacetime, let alone curved. Many physicists have worked on proving these conditions in free quantum field theories. It has been established that ANEC holds in Minkowski space for free scalar fields [[22], [23]], for Maxwell fields [23],and arbitrary two dimensional theories with positive energy and a mass gap [24].

During the last few years it has been understood that ANEC is not just a true statement for QFTs, but one of their fundamental properties. The latter has been understood from three different angles for interacting QFTs, in flat spacetime. In 2014 Kelly and Wall [25] proved ANEC for a class of strongly coupled conformal field theories using AdS/CFT. Later, in 2016, Faulkner, Leigh, Parrikar and Wang [26] proved ANEC from the point of view of quantum information and , in 2017, Hartman, Kundu and Tajdini [27] proved ANEC using causality, i.e. using that commutators should vanish at spacelike separation. We encourage the interested reader to look up these papers and also the lectures of Thomas Hartman for the Spring School on Superstring Theory and Related Topics 2018, that can be found here [28]. Finally, we must note that there are some proposals for curved spacetimes, but no actual proof.

2.4 Casimir effect

2.4.1 Electromagnetic Casimir effect

In 1948 Hendrik Casimir showed that in the presence of two conducting plates distorts the vacuum energy of the electromagnetic (EM) field [2]. In particular, it is found to be negative relative to the normal zero point energy.

This can be explained as follows. The plates are acting as boundaries. Thus, they are forcing the waves to be quantized due to the interactions between the atoms of the plates and the EM field. The plates are separated by distance L. So, the modes that have longer wavelength than L, will not be able to fit. This means that we are “missing” some modes between the plates. Thus, the vacuum energy we calculate is essentially lower than the vacuum energy of the Minkowski vacuum that contains all modes.

The explicit calculation of the electromagnetic Casimir stress tensor can be found in Appendix A. The result is the following:

T_Casimirµν = π 2 720a4      −1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 −3      (2.3)

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Let’s see which of the energy conditions are violated in the case of the EM Casimir effect. Since the energy density (ρ = Ttt_{) is negative the WEC is automatically violated. Moreover,}

it’s easy to show that NEC is also violated since ρ + pz < 0. The aforementioned violations

do not come as a surprise since Casimir is a quantum effect. Of course, in all the previous analysis we have assumed that we have perfectly conducting plates. If the plates are realistic their mass is always much larger than the Casimir energy density and the the averaged energy conditions are not violated.

In Visser’s book [29] there is an explicit analysis about what happens with the averaged energy conditions. In a nutshell, he defines another Casimir stress tensor, similar to (2.3), that corresponds to having realistic metal plates:

T_Casimirµν = σˆtµˆtν[δ(z) + δ(z − a)] + Θ(z)Θ(a − z) π

2

720a4 [ηµν− 4ˆz µ

ˆ

zν] , (2.4)

where ˆtµ_{is the unit vector in the time direction and the plates are not ideal and have surface}

mass density σ. Then he writes down ANEC and immediately infers that the only way it can be violated is when σ is physically unreasonable. So, it’s safe to say that ANEC is obeyed. The only case that some averaged energy conditions, like the AWEC, are violated is when a photon is travelling parallel to the plates (ANEC is still obeyed).

From the above discussion we see that the case of realistic plates does not seem very promising for our ultimate goal to built a traversable wormhole. However, Casimir effect may also arise from the topology of spacetime, for example if we impose periodic boundary conditions

2.4.2 Topological Casimir effect

As we mentioned, a variation of the original electromagnetic Casimir effect, is the Casimir effect that arises due to non-trivial topologies. For example, if we are at 1 + 1 flat spacetime and we take our universe to be periodic in the spatial direction (with period L), the Casimir stress tensor of some field will be non zero. Let’s consider the simplest example, the free massless scalar. The momentum of this field will be quantized in the spatial direction, due to the periodicity. The field modes are:

uk= (2Lω)−1/2ei(ωt−kx), (2.5)

where k = 2πn_L , n = 0, ±1, ±2, · · · . We want to calculate h0L|Tµν|0Li, where |0Li is the

vacuum of the quantum field in the periodic universe and |0i is the vacuum of the same quantum field in normal Minkowski spacetime. Of course, if we take L → ∞ then we should recover h0|Tµν|0i. The stress-energy tensor components of the scalar field in two dimensions are: Ttt = Txx = 1 2 ∂φ ∂t 2 +1 2 ∂φ ∂x 2 Ttx = Txt= ∂φ ∂t ∂φ ∂x. (2.6)

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The field can be expanded as: φ =X k akuk(t, x) + a † ku ∗ k(t, x) , (2.7) where ak/a †

k are the annihilators/creators and uk the field modes. Using (2.7) we find that:

∂µφ∂νφ = X k X k0 ak∂µuk+ a † k∂µu ∗ k a0_k∂µu0k+ a † k0∂µu0∗k = X k X k0 (2Lω)−1/2(2Lω0)−1/2kk0−akak0eikxeik 0_x + aka†_k0eikxe−ik 0_x + a†_kak0e−ikxeik 0_x − a†_ka†_k0e−ikxe−ik 0_x , (2.8)

where k is the momentum tensor. In order to go from the first to the second line we have used equation (2.5). We can now find the expectation value of ∂tφ∂tφ:

By performing a similar calculation we also calculate ∂xφ∂xφ =P_k |k|

2L. For a free massless

scalar in two dimensions we have ω = |k| and thus finally we can find the timelike component of the stress tensor to be:

Let’s regulate the sum by giving a penalty to the high frequency modes, using the Heat-Kernel cutoff: h0L|Ttt|0Li = X k |k| 2L = 2π L2 X n ne−a|k|= 2π L2 X n ne−2πaL , _(2.11)

where a is the regulator. In the end we are going to a → 0. For convenience, we define = 2πa_L and proceed.

h0L|Ttt|0Li = 2π L2 X n ne−n= 2π L2 X n ∂ ∂ e −n₌ 2π L2 ∂ ∂ X n e−n = 2π L2 ∂ ∂ 1 1 − e− = 2π L2 e− (1 − e−₎2 = 2π L2 1 2 − 1 12+ · · · = 1 2πa2 − π 6L2, (2.12)

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In order to finally obtain the Casimir stress tensor we have to subtract the expectation value of the stress tensor on the original Minkowski vacuum from the one on |0Li. Thus:

T_ttCasimir = h0L|Ttt|0Li − h0|Ttt|0i = h0L|Ttt|0Li − lim

L→∞h0L|Ttt|0Li = −

π

6L2 (2.13)

Thus, the scalar field has a non zero Casimir stress tensor due to the periodicity of spacetime. Of course similar effects exist in different dimensions.

3 Traversable wormholes from Casimir energy

3.1 An attempt to construct a wormhole using Casimir energy

The previous calculations have been in four dimensional Minkowski space. But one could wonder, if we perform a Weyl transformation to our metric and assume that the parallel planes we used before are the boundaries of AdS, could we get a metric that describes a wormhole? Of course, the generalized second law (GSL) of causal horizons states that it’s not possible to have traversable wormholes connecting two disconnected regions, but it is worth to try. In this scenario, we have:

gµν = Ω(z)2ηµν, (3.1)

where Ω(z)2 _{is the conformal factor. The variable z is goes from 0 to a. As, we mentioned}

before, a is the separation between the two ideal plates. Using the metric (3.1) we can calculate the components of the Einstein tensor:

Gtt = −Gxx = −Gyy = (Ω(z)0)2− 2Ω(z)Ω(z)00 Ω(z)2 Gzz = 3 (Ω(z)0)2 Ω(z)2 . (3.2)

Moreover, the stress-energy tensor transforms as [30]: ˜

Tµν = Ω(z)−2Tµν, (3.3)

where Tµν is the Casimir stress-energy tensor of the photon:

T_µνCasimir = π 2 720a4      −1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 −3      . (3.4)

We can now use (3.2) and (3.3) in order to write down the Einstein equations. Since we want our wormhole to be asymptotically AdS we need to also add a cosmological constant to the Einstein equations. So, we have:

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with Λ < 0. Equation (3.5) gives us two separate equations: (Ω(z)0)2 − 2Ω(z)Ω(z)00_{= −}8π3GN 720a4 − ΛΩ(z) 4 _(3.6) and (Ω(z)0)2 = −8π 3_G N 720a4 − ΛΩ(z) 4_. _(3.7)

We can rewrite the second one as: dΩ dz = ± r −ΛΩ4₋ GNc0 a4 , (3.8) where c0 ≡ 8π 3

720 and we have dropped the argument of Ω for convenience. We want our

space-time to be asymptotically AdS. So, near the boundaries the metric should asymptotically be:

ds2 = `

2

z2 −dt

2_{+ dz}2_{+ dx}2_{+ dy}2_, _{z > 0} _(3.9)

which covers half of AdS and is conformally equivalent to half-space Minkowski spacetime. As we see from (3.9), Ω should be infinite near the boundaries and take a minimum value in the center. So, if we want to integrate dz from 0 to a/2 (the center), we have to pick the negative sign in (3.8) since in this region Ω is decreasing:

− Z Ωmin ∞ dΩ q −ΛΩ4₋ GNc a4 = Z a/2 0 dz ⇒ a 2 = Z ∞ Ωmin dΩ q −ΛΩ4₋ GNc0 a4 , (3.10)

where we integrate If we make a coordinate change and define u ≡ Ωa the above equation takes the following form:

1 2 = Z ∞ aΩmin du √ −Λu4_{− G} Nc0 (3.11) As we can see, the a’s dropped. So, we see that in principle we want the r.h.s to be of order one. Then, we make a second change of coordinates and define x = −_GΛ

Nc0 1/4 u and thus (3.11) becomes: 1 2 = GNc0 Λ 1/4Z ∞ xmin dx √ x4_{− 1} (3.12)

where xmin = aΩmin

Λ GNc0

1/4

. The integral at the r.h.s of (3.12) is an number of order one. Thus, for (3.12) to be true we need GN ∼ Λ, which means that `P ∼ `. This is problematic

since we started by assuming that they were not of the same order. Moreover, the regime that we know how to handle is when ` `P, since for distances of Planck scale order we

expect quantum gravity effects to appear, and we do not yet know how to treat them. Thus, this does not seem to be the best approach in order to build a traversable wormhole from Casimir energy.

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3.2 Casimir energy of a long wormhole throat

As we saw, Casimir energy can arise without the use of actual plates but due to the topology of spacetime. So, it seems like the topological Casimir effect might be a better candidate for the construction of traversable wormholes. This is exactly what is discussed by Luke Butcher in his paper “Casimir Energy of a Long Wormhole Throat” [3].

The idea is whether a wormhole, itself, can produce the Casimir energy it requires. In order to achieve this the shape of the wormhole has to be optimized in a way as to produce as much negative energy density as possible and require as little negative energy density as possible. In order to achieve this we have to make the wormhole much longer that it is wide. We start with a static spherically symmetric metric that can describe a traversable wormhole:

ds2 = −dt2+ dz2+ A dθ2+ sin2θdφ2 , A =√L2_{+ z}2_{− L + a,} _(3.13)

where 2L is the length of the wormhole throat and a its radius. As we can see in figure 3.2 this metric represents a surgically constructed wormhole that connects two flat regions.

The stress tensor for this metric in the orthonormal basis is: Tµˆˆν = Gµˆˆν κ = L2 (L2_{+ z}2_)A2_κdiag 1, −1,√ A L2_{+ z}2, A √ L2_{+ z}2 + 2L 2 (L2_{+ z}2₎3/2_Aκdiag (−1, 0, 0, 0) (3.14) We are going to assume that L ≥ a and consequently √ A

L2_+z2 ≤ 1. If that’s the case then it’s

straightforward to see that the fist part of the stress tensor obeys all the energy conditions, whereas the second does not. So, essentially the second part is the “exotic” matter that we need in order to support the wormhole. Also, we can see that the second part of the stress tensor takes its maximum value at z = 0 (in the center of the wormhole), and this value is :

ρmax_required ∼ 2

Laκ, (3.15)

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We would also like to know how much negative energy can be produced. As we saw, in the center of the wormhole we need the most negative energy. We can find the radius of the wormhole near the center if we Taylor expand A around z = 0. The result is:

A = a = z 2 2L+ O z4 L3 , (3.16)

and if L is very large comparing to z then A ∼ a. So, if there is a field in our spacetime it will become quantized inside the wormhole throat. The field modes that have wavelength shorter than L will fit in the wormhole throat, whereas the ones with longer wavelength will not. Due to this, we expect Casimir energy to be produced, which will be of order:

ρmax_produced ∼ ~

a4. (3.17)

It’s easy to see that if we keep a constant and make L very big we minimize the required energy and make the produced energy approximately constant. Now, if ρmax

required ≈ ρmaxproduced

we have that: 2 Laκ ∼ ~ a4 ⇒ a ∼ (`p) 2 L a, (3.18)

from which we can infer that if L a, both of L and a are bigger than the `p. So, by taking

L → ∞ we have the following metric to work with:

ds2 = −dt2 + dz2+ a dθ2+ sin2θdφ2 . (3.19) We are going to consider a non-minimally coupled massive scalar field in our spacetime, that has the following action:

S = Z

dx4√−g (∇ϕ)2_{+ (m}2_{+ ξR)ϕ}2_, _(3.20)

where ξ is the coupling constant for the interaction term between gravity and the scalar field. For the case of the conformally coupled scalar field ξ = 1₆. The classical stress tensor can be calculated by varying the action with respect to the metric:

Tµν = 2 √ −g δS δgµν = ∇µϕ∇νδphi+ξ Rµνϕ 2_{− ∇} µ∇ν(ϕ2)−gµν 1 − 4ξ 2 (∇ϕ) 2_{+ (m}2_{+ ξR)ϕ}2 (3.21) The field will become quantized inside the wormhole throat, so we will expand it as follows:

φ =X n ϕ−_na−_n + ϕ+_na+_n , (3.22) where a+ n = (a − n) †

are the creation/annihilation operators, satisfying the usual commutation relations. As we saw in section 2.4.2, the goal is to calculate the difference between the expectation value of the stress tensor of ϕ in the spacetime with the non-trivial topology (here the wormhole throat) and the expectation value of the stress tensor of ϕ in normal Minkowski spacetime, i.e.:

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where |0i is the Minkowski vacuum and |0ai is the “throat” vacuum. The actual calculation

is a very extensive and involved one and goes beyond the scope of this analysis. We are going to give the final result and discuss it. The Casimir stress tensor is:

T_µˆ_ˆCasimir_ν = 1 2880π2_a4 diag (−1, 1, −1, −1) 2 log a a0 + diag (0, 0, 1, 1) , (3.24)

where a0 is length scale introduced by the regularization scheme the author chose. However,

it has a simple meaning. It is clear from (3.24) that if our wormhole has radius a0 the

Casimir energy density becomes zero. Thus, we can interpret it as the radius that the Casimir energy density vanishes. If a is sufficiently larger than a0 then the Casimir energy

density is negative. Consequently, the dominant and weak energy condition are immediately violated because Tˆ_0ˆ₀ < 0. Moreover, the required energy density ρmax_required, that we previously

defined, can be supplied by the stress tensor of the scalar field. So, we may write: 2

Laκ =

log (a/a0)

1440π2_a4 ⇒ a 2

= `2_p(L/a)log (a/a0)

360π . (3.25)

If L a, indeed the wormhole has a macroscopic throat-radius a `p. However, up to this

point we have completely ignored the non-“exotic” part of the stress tensor.

We saw before that for a > a0the weak and dominant energy conditions are automatically

violated since the energy density is negative. Let’s now check what happens with the null energy condition. We need that Tµˆˆνkµˆkνˆ < 0, for some null vector kµˆ. So, we have:

Tµνkµkν = T00k0k0+ T11k1k1+ T22k2k2 + T33k3k3 < 0, (3.26)

where we have dropped the hats for convenience. We also know that in order for kµˆ _{to be a}

null vector it needs to satisfy:

kµkµ= 0 ⇒ −k02+ k 2 1 + k 2 2 + k 2 3 = 0 ⇒ k 2 3 = k 2 0 − k 2 1 − k 2 2. (3.27)

By using (3.27) and also that T11= −T00 and T22= T33, (3.26) becomes:

k2₀− k2

1 (T00+ T22) < 0. (3.28)

We substitute T22 and T00 and get:

k₀2− k2₁ 1

2880π2_a4(−4 log (a/a0) + 1) < 0 (3.29)

From (3.27) we know that k2₀− k2

1 = k22+ k32 > 0 and thus −4 log (a/a0) + 1 should be smaller

than zero. This happens when:

a > a0e1/4 (3.30)

Thus, the NEC is violated if a > a0e1/4 for all null vectors except for k0 = ±k1, i.e for

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wormhole will “see” negative energy except the null ray that is travelling directly parallel to the throat

This particular null direction is the one causing problem with the stability of the worm-hole. We would like to be able to solve Einstein’s equations, but with some additional ordinary matter3_{. Then we have:}

Gµν = 8πκ TµνCasimir+ T ordinary

µν , (3.31)

and let’s contract this with kµ_{, a null or timelike vector as follows:}

Gµνkµkν = 8πκ TµνCasimirkµkν + Tµνordinarykµkν . (3.32)

For almost all null and timelike vectors it’s true that T_µνCasimirkµkν < 0 , so we should be able to accommodate negative values for Gµνkµkν. However, for kµ = (1, ±1, 0, 0) we have that

T_µνCasimirkµkν = 0 and thus if Gµνkµkν < 0 then also Tµνordinarykµkν < 0, which is a contradiction

since ordinary matter satisfies the null energy condition by definition. Consequently, it is not possible to solve the Einstein equations for this kind of wormhole just by using the Casimir stress energy tensor and some ordinary matter and the Casimir energy produced by the wormhole itself is not enough to stabilize it permanently. The reason behind this it that the wormhole throat is spherically symmetric which makes the Casimir stress tensor have the form TCasimir = diag (ρ, −ρ, p, p). Thus, when contracted with kµ= (1, ±1, 0, 0) it gives zero. In order to avoid this effect, the author suggests inducing some symmetry breaking, for example by “twisting” the throat.

However, even though the wormhole is not permanently stable it collapses slowly allowing a null ray to cross it as is shown explicitely in [3]. So, this is an example where Casimir energy allows for the creation of a traversable wormhole. Of course, it is slowly collapsing and is not stable, but it is traversable nonetheless.

4 BTZ black hole

1+2-dimensional gravity, at first sight, looks trivial. In particular general relativity has no Newtonian limit and the graviton has no propagating degrees of freedom. So, it came as a surprise when Baados, Teitelboim and Zanelli discovered the BTZ black hole solution [31].

The BTZ black hole differs from the Kerr and Schwartzchild solutions in some important aspects. Firstly it is asymptotically AdS, instead of asymptotically flat and secondly it does not have a curvature singularity at the origin. However, it is indeed a black hole with a horizon, it appears as the final state of collapsing matter, and it has thermodynamic properties similar to these of the 1+3-dimensional black hole.

The uncharged, non-rotating BTZ black hole metric in “Schwartzchild” coordinates is: ds2 = −r 2_{− r}2 h `2 dt 2₊ ` 2 r2_{− r}2 h dr2 + r2dφ2, (4.1)

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where rh is the horizon radius and ` is the radius of AdS. The φ coordinate has period 2π,

the mass of the black hole is M = r2h

8GN`2 and its inverse temperature is β =

2π`2

rh . In some

cases it will be more convenient to use the metric in Kruskal coordinates, which smoothly cover the maximally extended two-sided geometry. In Kruskal coordinates the metric has the following form:

ds2 = 4` 2_{dudv + r}2 h(1 − uv) 2 dφ2 (uv + 1)2 , (4.2)

where u > 0 and v < 0 in the right wedge (see figure below 2), uv = −1 at the boundaries and uv = 1 at the singularities.

u v

L R

Figure 2: On the right/left we see the Kruskal/Penrose diagram of the maximally extended BTZ black hole.

As we see from figure 2, the maximally extended BTZ black hole has two asymptotically AdS regions (L, R), that are connected by a non-traversable wormhole. That wormhole collapses into a singularity in the future and in the past. The crossing lines represent the horizons and separate the spacetime into four regions. The left and right regions are the exterior of the black hole. The upper and lower regions are the future and past interior respectively.

4.1 BTZ propagators

This discussion is based on [32]. In order to obtain the bulk-to-boundary propagator for the BTZ black hole, we can exploit the fact that it is a quotient of AdS3. So, we only need to

add a sum over images on the bulk-to-boundary propagator of AdS3, in order to obtain the

BTZ propagators.

Let’s start from the bulk-to-boundary propagator. We need to specify a “source” point on the boundary b0 and a “sink” point in the bulk x. For a scalar field of mass m the

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bulk-to-boundary propagator in the right wedge, up to normalization, is: K(x, b0)RR ∼ ∞ X n=−∞ − s r2_{− r}2 h r2 h cosh (rh(t − t0)) + r rh cosh rh(φ − φ0 + 2πn) !−∆ , (4.3)

where ∆ is the conformal dimension of the boundary operator dual to the massive scalar field. Equation (4.3) is valid whenever the source and sink points are in the same region. If we want the sink point to be in a the left wedge then we replace t → t − iβ₂. Then we have:

K(x, b0)LR ∼ ∞ X n=−∞ − s r2_{− r}2 h r2 h cosh rh t − t0 −iβ 2 + r rh cosh rh(φ − φ0+ 2πn) !−∆ , (4.4) As we saw before β = 2π`_r 2

h . Assuming ` = 1, the propagator K(x, b

0₎

LR takes the form:

K(x, b0)LR ∼ ∞ X n=−∞ s r2_{− r}2 h r2 h cosh rh(t − t0) + r rh cosh rh(φ − φ0+ 2πn) !−∆ , (4.5)

We notice that (4.3) can be singular, whereas (4.5) is always finite. The reason for that is that in the first case the points are timelike separated, whereas in the second they are spacelike separated.

The boundary-to-boundary propagator can be acquired by sending the bulk point x to the boundary. This is done in the following way:

P (b, b0) ∼ lim

r→∞r

∆_{K(x, b}0

). (4.6)

By doing the above, we obtain the boundary-to-boundary propagators:

P (b, b0)RR ∼ ∞ X n=−∞ (− cosh rh(t − t0) + cosh rh(φ − φ0+ 2πn)) −∆ (4.7) P (b, b0)LR ∼ ∞ X n=−∞ (cosh rh(t + t0) + cosh rh(φ − φ0+ 2πn)) −∆ (4.8) In (4.8),we have assumed that in the second copy of the CFT the time increases towards the future.

4.2 The thermofield double formalism

This discussion follows from [33] and [34].

The thermofield double formalism, developed by Takahashi and Umezawa [35], is a trick we use to treat the thermal, mixed state ρ = e−βH as a pure state in a bigger system. If we have a QFT with some Hamiltonian H, we first double the degrees of freedom by considering two copies of this QFT. The states of this doubled QFT are |ni1|mi2. These two QFTs live

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in different spacetimes and are non-interacting. Now, in this doubled system we consider a particular pure state, i.e. the thermofield double state:

|T F Di = 1 pZ(β) X n e−βEn/2_|ni 1|ni2, (4.9)

where the 1, 2 indicates the Hilbert space where the state is defined, Z(β) is the partition function of one copy of the QFT with inverse temperature β. The density matrix of the doubled QFT in this state is:

ρtot = |T F DihT F D| (4.10)

The reduced density matrix of the first system is:

ρ1 =Tr2ρtot = X 2 2hm| 1 Z(β) X n,n0 e−βEn/2_|ni 1|ni2 2hn0|1hn0|e−βE 0_/2 ! |mi2 = X n e−βEn/2_|ni 1 1hn| = e−βH1 (4.11)

So, this pure state in the double system cannot be distinguished from a thermal state. For the Hamiltonian of the doubled system we have two options. Either ˜Htot = H1+ H2

or Htot = H1− H2. We shall choose Htot, under which |T F Di is time independent since the

phases cancel.

We saw in the previous chapter that an eternal BTZ black hole has two asymptotic boundaries. It has been proposed by Maldacena [34], that the BTZ is dual to two copies of a CFT, in the thermofield double state. This was shown by performing the path integral on the boundary CFT. We must note that even though the two CFTs are not interacting the expectation value of two operators, each from one of the two independent CFTs is non zero. This is due to the entanglement of the two theories, or equivalently, due to the presence of the wormhole, as Maldacena and Susskind proposed [36]. Moreover, the Hamiltonian we chose before, Htot, is dual to the Hamiltonian that generates the time evolution along the

isometry ∂t in the bulk.

5 BTZ shock-waves

In this chapter, we are going to mainly review the work of Shenker and Stanford on shock waves in the BTZ black hole [37], [38]. Previously, we explored the case of an unperturbed BTZ black hole and some of its properties. It is interesting to see what happens when we mildly perturb it. So, starting form the thermofield double state we will add a perturbation to it and see what happens. We can do this by adding some particle at the left boundary, at some time t. Thus, we consider a CFT state of the form:

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where W (tw) is a local operator that acts unitarily on the left CFT and raises the energy by

an amount E. We assume that the energy is much smaller than the mass of the black hole. There are at least two ways of thinking of that state. The first one is to think of it as prepared by changing the Hamiltonian at some time, which means that we start from the thermofield double state and then we perturb it at time tw. This scenario is depicted in subfigure (a) of

figure 3. The second way is to think of it as a state with a time independent Hamiltonian, in which case we have to follow this perturbation backwards through the past horizon. So, in the second scenario, which is shown in subfigure (b) of figure 3, the perturbation comes out of the white hole, approaches the left boundary at tw and falls in the black hole.

tw L R (a) tw L R (b)

Figure 3: In this figure we see the insertion of particles at the left boundary at some early time. The perturbation falls in through the future horizon. The double blue line

Naively, we would think that this perturbation will not have any effect on the geometry. However, we may instead release the perturbation from the left boundary long in the past, as in figure 4 . As we know, translation in Killing time acts as a boost at the near horizon region. So, at the local frame of the timeslice t = 0, the energy we are going to measure is going to be:

Ep ∼

E` R e

rhtw/`2_, _(5.2)

where E is the initial energy of the particle. Thus, in this frame the particle is actually a high energy shock wave that has a back reaction on the geometry. For simplicity, we consider a spherically symmetric null shell of matter. The resulting geometry is obtained by gluing two BTZ black holes of mass M and M + E accross the null surface vw = e−rhtw/`

2

. We will use u, v coordinates for the past of the shell and ˜u, ˜v for its future. Since we are “tossing” a positive energy object in our black hole we increase the mass and that makes the radius grow. So, using that M = r2h

8GN`2, the new radius is:

˜ rh =

r

M + E

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If we take a look at the metric (4.2), we see that the following has to hold: ˜ rh 1 − ˜u ˜vw 1 + ˜u ˜vw = rh 1 − uvw 1 + uvw (5.4) tw L R

Figure 4: Here, we release the perturbation very early.

In order to solve (5.4) we will assume that ˜vw = vw and we will define the new variables

x = uvw and ˜x = ˜u˜vw. Then (5.4) becomes:

˜ rh rh 1 − ˜x 1 − x = 1 + ˜x 1 + x ⇒ 1 + E 2M 1 − ˜x + x − x 1 − x = 1 + ˜x + x − x 1 + x ⇒ 1 + E 2M 1 + x − ˜x 1 − x = 1 + x − x˜ 1 + x , (5.5)

where we have Taylor expanded (5.3) around _ME = 0, and we have added and subtracted x both in the left and right hand side terms, in order to go from the second line to the third. Next, we add to both sides the term − 1 + x−˜_1−xx and get:

E 2M 1 − ˜x 1 − x = 2(˜x − x) (1 − x)(1 + x) ⇒ ˜ x − x 1 + x = E 4M(1 − ˜x). (5.6) Then, we substitute x, ˜x: vwu − v˜ wu 1 + vwu = E 4M(1 − vwu) ⇒˜ ˜ u − u v−1 w + u = E 4M(1 − vwu) ⇒˜ ˜ u − u = E 4M(1 − vwu)(v˜ −1 w + u) ⇒ ˜u = u + _4ME (v_w−1+ u) 1 + _4ME (1 + vwu) ⇒ ˜ u = u + E 4Mv −1 w − E 4Mvwu 2_{+ O} E2 M2 , (5.7)

where we have expanded once again around _ME = 0 in order to from the second line to the third. Since vw = e−rhtw/`

2

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going to be approximately zero and we can ignore it. Consequently, the solution is a simple shift in the v coordinated, namely:

˜

u = u + a, a = E 4Me

rhtw/`2_, _(5.8)

where we have substituted vw in the last step of(5.7).

Figure 5: Here we see the (non-square) Penrose and Kruskal diagrams of the perturbed BTZ black hole. The red parallel lines represent the shock wave. The horizons now are not touching any more. They have separated by an amount of a.

We can write our new metric (after the backreaction) as: ds2 = 4` 2_{dudv + r}2 h[1 − ((u + aθ(v)) v] 2 dφ2 [1 + (u + aθ(v)) v]2 , (5.9)

where θ(v) is the step function. What this means is that if we send a signal from the right boundary towards the left boundary, it will suffer a time delay when it reaches the horizon v = 0.

tw

L R

Figure 6: Here is the square Penrose diagram of the perturbed BTZ black hole. The double red lines represent the shock wave. The orange line represents a signal that we send from the right boundary.

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As we see in figure 6, when the signal collides with the shock wave, it suffers a time delay and essentially ends up in the singularity. So, for our purpose of constructing a traversable wormhole, one could say that the shock wave make it even harder for a signal to cross to the other side. In order for a signal to avoid falling in the singularity we would like to have exactly the opposite effect, namely, our signal to gain a time advance instead of a time delay. If we had an operator that could create a negative energy shock wave instead of a positive one, the shift a would be negative. In this case, the signal would meet with the negative energy shock wave, shift towards the opposite direction and reappear on the left boundary. Below, we depict how such a hypothetical configuration would look.

tw

L R

6 Non-local coupling in 1+1 flat spacetime

In this chapter, we are now going to explore a simple way of getting negative energy density in QFT. As we will see, if we add in the action of our system a term of the form δS = gφLφR

at a certain time, the resulting expectation value of the stress tensor consists of positive and negative energy shock waves.

6.1 First order calculation

We consider a free massless scalar field in two dimensions, whose action is: S = −

Z ₁ 2∂µφ∂

µ

φ. (6.1)

We will deform the system by adding an interaction term of this form:

δS = gφLφR (6.2)

where φL,Ris the field operator φ evaluated at xLand xRrespectively, at a time-slice t = 0, in

two dimensional Minkowski space. We assume that the two points are spacelike separated. For convenience, we will use light-cone coordinates. We are now going to calculate the expectation value of the normal ordered stress-energy tensor on the state |Ψi = eigφLφR_|0i,

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to first order in g:

hΨ| : Tuu(u) : |Ψi =e−igφLφR : ∂uφ∂uφ : eigφLφR = h(1 − igφLφR) : ∂uφ∂uφ : (1 + igφLφR)i =

ig h: ∂uφ∂uφ : φLφRi − ig hφLφR : ∂uφ∂uφ :i = ig h[: ∂uφ∂uφ :, φLφR]i .

(6.3) We define C ≡ h: ∂uφ∂uφ : φLφRi. Then assuming φR, φL and ∂uφ are Hermitian, we can

recover the the commutator h[: ∂uφ∂uφ :, φLφR]i by taking the imaginary part of C:

hΨ| : Tuu(u) : |Ψi = −2gIm (h: ∂uφ∂uφ : φLφRi) = −4gIm (h∂uφφLi h∂uφφRi) , (6.4)

where in order to go from the second to the third equality we have performed the Wick contraction. The correlators we need to calculate are non-time ordered. Therefore, we are going to use the Wightman function rather than the Feynman propagator. The Wightman function for the free massless scalar in two dimensions is [39]:

W (t, x; t0, x0) = hφ(t, x)φ(t0, x0)i = − 1

4π[log [iµ (∆t + ∆x − i)] + log [iµ (∆t − ∆x − i)]] , (6.5) where µ is an infrared cutoff. In lightcone coordinates it takes the following form:

W (u, v; u0, v0) = hφ(u, v)φ(u0, v0)i = − 1

4π[log [iµ (∆u − i)] + log [iµ (∆v − i)]] . (6.6) Then by using (6.6), we may compute (6.3) to be:

hΨ| : Tuu(u) : |Ψi = − 4g 16π2Im 1 (u − uL) − i · 1 (u − uR) − i = − g 4π2 (u − uL) ((u − uL)2+ 2) · ((u − uR)2 + 2) − g 4π2 (u − uR) ((u − uR)2+ 2) · ((u − uL)2+ 2) , (6.7)

and if we take the limit → 0 we finally obtain: hΨ| : Tuu(u) : |Ψi = − g 4π δ(u − uR) u − uL +δ(u − uL) u − uR , (6.8)

where we have used that:

δ(x) = 1 π lim→0

x2₊2. (6.9)

Following exactly the same procedure we find that hΨ| : Tvv(v) : |Ψi = is:

hΨ| : Tvv(v) : |Ψi = − g 4π δ(v − v_R) v − vL + δ(v − vL) v − vR . (6.10)

Hence, the resulting configuration of the energy density4 _is:

4_{The energy density is defined as ρ ≡ T}

00(t, x), which in the case of the free massless scalar in 2d is equal

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v u

(uL, vL) (0, 0) (uR, vR)

Figure 7: The blue/red lines represent the regions of space where we have negative/positive energy density.

A light ray travelling along u = 0 will only “pass through” negative energy density and we will thus have R∞

−∞hTuui du < 0.

Up to this point, everything we have calculated is to first order in g. We would rather like to calculate the expectation value of the components of the stress tensor up to second order in g,

hΨ| : Tuu(u) : |Ψi =e−igφLφR : ∂uφ∂uφ : φ eigφLφR =

1 − igφLφR− g2_φ LφRgφLφR 2 : ∂uφ∂uφ : 1 + igφLφR− g2_φ LφRgφLφR 2 = − 4gIm (h∂uφφLi h∂uφφRi) + g2hφLφR : ∂uφ∂uφ : φLφRi − g2 2 hφLφRφLφR: ∂uφ∂uφ :i − g2 2 h: ∂uφ∂uφ : φLφRφLφRi , (6.11) where we have omitted terms of higher order than g2_{. The final result is (for details of the}

calculation, see Appendix B):

hTuui = −4gIm (h∂uφφLi h∂uφφRi) +

g2(−2 hφLφRi hφL∂uφi hφR∂uφi − 2 hφLφRi h∂uφφLi h∂uφφRi + 2 hφLφRi hφL∂uφi h∂uφφRi

+2 hφLφRi hφR∂uφi h∂uφφLi + 2 hφLφLi hφR∂uφi h∂uφφRi − hφLφLi hφR∂uφi hφR∂uφi

− hφLφLi h∂uφφRi h∂uφφRi + 2 hφRφRi hφL∂uφi h∂uφφLi − hφRφRi hφL∂uφi hφL∂uφi

− hφRφRi h∂uφφLi h∂uφφLi)

(6.12) Similarly, we can compute the expectation value of the Tvv component of the stress tensor.

It has exactly the same form with (6.12), except u → v. Both of hTuui and hTvvi, to second

order in g, contain correlators of the form hφLφLi and hφRφRi, which diverge. In order to

cure this and get some meaningful results we need to smear our sources. This means that we will “spread” our source in time and space. Since we are working in lightcone coordinates, a convenient choice is to make our sources diamond-shaped.

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6.2 Smearing of the sources

From now on instead of φL and φR, we are going to use:

OL≡

Z vL+A

vL−A

Z uL+A

uL−A

dvdu φ(u, v), and OR ≡

Z vR+A

vR−A

Z uR+A

uR−A

dvdu φ(u, v), (6.13) where 2A is the side of the diamond (see figure 8). Now, we have to compute all the smeared two-point functions. For demonstration purposes we will perform one of the calculations :

h∂uφ(u, v)OLi = 1 4A2 Z vL+A vL−A Z uL+A uL−A dv0du0h∂uφ(u, v)φ(u0, v0)i = − 1 16πA2 Z vL+A vL−A Z uL+A uL−A dv0du0 1 u − u0_{− i} = − 1

8πA(log (uL− A − u + i) − log (uL+ A − u + i)) ,

(6.14)

where we have used (6.6), and we have divided by the volume of the diamond, 4A2, in order to have the correct dimensions.

Next, we will take the limit → 0. We need to be careful and divide the space into different zones, since the real part of the logarithm arguments can be either negative or positive, depending on where we are.

I II III IV V

(0, 0)

uL uR

Figure 8: The gray diamond areas are the smeared sources.

Zone I (u < uL− A) :

h∂uφ(u, v)OLiI= −

1

8πA→0lim+(log (uL− A − u + i) − log (uL+ A − u + i)) =

− 1 8πAlog u_L− A − u uL+ A − u . (6.15) Zone II (uL− A < u < uL+ A) : h∂uφ(u, v)OLiII= − 1

− 1 8πA iπ + log −uL− A − u uL+ A − u , (6.16) Zone III − V (u > uL+ A) :

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h∂uφ(u, v)OLi_III−V = −

1

− 1 8πAlog uL− A − u uL+ A − u . (6.17)

Finally, we can repackage everything as: h∂uφ(u, v)OLi = − 1 8πA iπ θ (u − uL+ A) θ (uL+ A − u) + log uL− A − u uL+ A − u . (6.18) In a similar fashion, we can also calculate the rest of the correlators appearing in (6.12) :

hOL∂uφ(u, v)i = − 1 8πA −iπ θ (u − uL+ A) θ (uL+ A − u) + log uL− A − u uL+ A − u , (6.19) h∂uφ(u, v)ORi = − 1 8πA iπ θ (u − uR+ A) θ (uR+ A − u) + log uR− A − u uR+ A − u , (6.20) hOR∂uφ(u, v)i = − 1 8πA −iπ θ (u − uR+ A) θ (uR+ A − u) + log uR− A − u uR+ A − u , (6.21) hOLORi = − 1 16πA −12A 2_{− 2(u} L− uR)2log uR− uL A + 2A + uL− uR A 2 log −2A − uL+ uR A + (2A − uL+ uR) log 2A − u_L+ uR A + 8A2log (Aµ) , (6.22) and hOLOLi = hORORi = − 1 2πlog (2Aµ) . (6.23)

Similarly, we find the associated two-point functions for hTvvi. The energy density is the sum

of hTuui and hTvvi and now that we have all the two-point functions at hand we can finally

calculate the energy density up to second order in g, with smeared sources. The resulting configuration is in the figure below:

a b c d

e

f g _h _i

(0, 0)

uL uR

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In the areas a, c, e, g, i the stress-energy tensor is zero. In the strip b the stress-energy tensor is: hTuuib = − g 32A2_πlog u_R− u − A uR− u + A − g 2

32A2_πlog (2Aµ) , uL− A < u < uL+ A (6.24)

In the strip f we have: hTuui_f = − g 32A2_πlog uL− u − A uL− u + A − g 2

32A2_πlog (2Aµ) , uR− A < u < uR+ A (6.25)

In the strip h: hTvvi_h = − g 32A2_πlog vL− v − A vL− v + A − g 2

32A2_π log (2Aµ) , vR− A < u < vR+ A (6.26)

and finally in d: hTvvi_d = − g 32A2_πlog vR− v − A vR− v + A − g 2

32A2_π log (2Aµ) , vL− A < u < vL+ A (6.27)

From the above results we notice that the O(g) term of the stress-energy tensor depends on the distance between the sources, as well as the side of the diamond. However, the O(g2₎

term only depends on the side of the diamond and the IR cutoff.

In order to see the plot of the stress tensor against the u coordinate, we will pick some values for uR, uL and A. So, for uR= −uL= 10 and A = 0.1, we plot hTuui in the strip b:

We notice that the stress tensor is almost linear in u inside the strip. Moreover, we see that it is slightly increasing with u. Next, we plot hTuui in the strip f , where we know that it is

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In strip f the stress tensor decreases as u increases. As we previously mentioned, this stress tensor violates the ANEC.

However, there are other bounds to the amount of negative energy we are allowed to have, that are commonly referred to as the quantum inequalities (QIs). In the next subsection we will see what the QIs are and check whether or not the stress tensor that we calculated obeys them.

6.3 Quantum Inequalities

In section 2.2,we saw, that there are two different approaches in order to find constraints analogous to the pointwise energy conditions. One approach is the average versions of the energy conditions. The second approach, which we are going to discuss here is the QIs, first introduced by Ford.[20]. Since then, they have been proved and refined by Ford, as well as others. The first versions of the QIs were constraints on the magnitude and duration of the negative energy fluxes and densities, measured by an inertial observer [40]. They resembled the uncertainty principle because they said that a pulse of negative energy cannot be arbitrarily intense for an arbitrarily long time. More precisely, they say that the duration of a negative energy pulse is inversely related to its magnitude. Later the QIs were also proved for the expectation value of the energy density in arbitrary quantum states, in d-dimensional Minkowski spacetime. Let’s assume we have:

ρ(t) = hTtt(t)i , (6.28)

the expectation value of the timelike component of the stress tensor evaluated on some arbitrary state, at time t. Then the QI for a massless field has the following form:

Z

dtρ(t)g(t, τ ) ≥ −C

τd, (6.29)

where d is the spacetime dimension. Moreover g(t, τ ) is a sampling/test function and C is a positive constant. The meaning of (6.29) is that if a negative energy pulse lasts for time of

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order τ , then its magnitude is bounded by −_τCd. The version that we are mostly interested

in this chapter is the QIs in two dimensional Minkwoski space [41]. In two dimensions for any light-ray travelling along v = constant, the QI has the following form:

Z du hTuu(u)i ρ(u) ≥ − 1 48π Z du(ρ(u) 0₎2 ρ(u) , (6.30)

where ρ(u) is a smooth, peaked smearing function that integrates to one (the sampling function we mentioned before). We would like to know if the QIs still hold when we add non-local couplings in our theory. Thus, we will use the stress tensor we previously derived (6.25) and we will choose the Gaussian as our smearing function.

Now, let’s assume a light-ray is travelling along v = 0. It will only “pass through” the negative strip of hTuu(u)i. Thus, when we integrate along the u direction we will only have

non-zero hTuu(u)i in the strip f . Then we have:

Z uR+A uR−A hTuu(u)i 1 σ√2πe −1 2( u−µ σ ) 2 ≥ − 1 48π 1 σ2. (6.31)

These inequalities are supposed to hold for any smearing function with the aforementioned attributes, and for any parameters. We immediately see that at the large σ limit the inequal-ity is violated. If we bring everything to the left-hand side and forget about the constants, we will have: σ Z uR+A uR−A du hTuu(u)i e −1 2( u−µ σ ) 2 ≥ −1 (6.32)

At the large σ limit the exponential will be approximately equal to one. So, (6.32) takes the form: σ Z uR+A uR−A du hTuu(u)i ≥ −1 (6.33) Since RuR+A

uR−A du hTuu(u)i < 0 the left-hand side of (6.33) can be arbitrarily negative. Thus,

we may conclude that when we add non-local sources in flat space, the quantum inequalities are no longer true.

As we saw, the addition of non-local coupling induces the violation of both of the available ways to constrain the negative energy density, ie. averaged version of energy conditions, and the QIs.

7 Non-local couplings in black holes

7.1 BTZ with smeared sources

This idea of the non-local couplings was first applied by Gao, Jafferis and Wall (GJW) in the case of the BTZ black hole [4]. We are going to review their results briefly, without demonstrating the details of the calculation. Starting with a maximally extended BTZ black

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hole, at some time t0 they coupled the two asymptotic boundaries of the black hole by adding

to the action a term of the form: δS = −

Z

dtdφ h(t, φ)OR(t, φ)OL(−t, φ), (7.1)

where O is a scalar primary operator with scaling dimension ∆, dual to a scalar field ϕ. As we can see the operators have been smeared over time and angle. By doing some dimensional analysis we can find some condition on the scaling dimension of O:

[dtdφ] + [h] + 2∆ = [S] ⇒ −2 + [h] + 2∆ = 0, (7.2)

where have expressed everything in units of energy. From (7.2), we can infer that in order for h to be bigger than zero, ∆ has to be smaller than one.

The ultimate goal of [4] is to calculate the expectation value of the stress tensor due to the insertion of these operators OL, OR. In order to do so they first calculate the modified

bulk-to-bulk propagator in the right wedge and then use the point-splitting method to compute the stress tensor, on the horizon V = 0.

The modified bulk-to-bulk two-point function in the right wedge in Kruskal coordinates is: Gh =C0 2π β 2∆−2 rh Z _dU 1 U1 dφ1 h(U1, φ1) 1 + U0V0 (U0_U 1− V0/U1) + (1 − U0V0) cosh rh(φ0− φ1) ∆ × 1 + U V (U/U1− V U1) − (1 − U V ) cosh rh(φ − φ1) ∆ + (U, φ ←→ U0, φ0), (7.3) where C0 = r_h2−2∆sin π∆ 2(2∆_π)2 2π β 2−2∆ and h(U1, φ1) = h 2π β 2−2∆

. The authors have also set the radius of AdS to one. For simplicity, we can take φ = φ0 and also set V = V0 = 0, which means that they calculate the two-point function on the horizon. By doing the aforementioned simplifications we end up with:

Gh = hC0 Z U U0 dU1 U1 Z U U1 1 2dy py2_{− 1} 1 U0_U 1+ y ∆ U1 U − U1y ∆ | {z } F (U,U0₎ + (U ←→ U0) | {z } F (U0_{,U )} , (7.4)

where y = cosh rh(φ1 − φ). Then, the stress tensor is then acquired by point splitting:

hTU Ui = lim U0_→U∂U∂U0(F (U, U 0 ) + F (U0, U )) = 2 lim U0_→U∂U∂U0F (U, U 0 ). (7.5)

The authors obtain the stress tensor numerically. In figure 10 we can see the stress tensor against the coordinate U .

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Figure 10: On the left we see the stress tensor against U in the case where we turn on the coupling at U0 = 1 and never turn it off. In the right sub-figure we see the stress tensor against U in the

case where we turn on the coupling at U0 = 1 and turn it off at Uf = 2. The coupling constant h

is assumed to be 1.

The stress tensor for ∆ < 1/2 is finite, but for ∆ > 1/2 it is divergent at the point where we turn on/off the coupling. However, this divergence is not important because it is integrable. Moreover, in sub-figure (b) of figure 10 we see that after the turning off of the coupling the stress tensor becomes positive. However, we need not worry since the relevant quantity in order to detect whether or not we have a traversable wormhole is the integral of the stress tensor along a null path. However, even though the stress tensor eventually becomes positive, its integral is always negative. The integrated stress tensor is:

Z ∞ U0 TU UdU = − hΓ(2∆ + 1)2 24∆_{(2∆ + 1)Γ(∆)}2_{Γ(∆ + 1)}2_` 2F1 1 2 + ∆, 1 2 − ∆; 3 2 + ∆; 1 1+U2 0 (1 + U0)∆+1/2 . (7.6) As we can see from figure 11, the integral of the stress tensor is always negative. Even when the stress tensor becomes positive after the turn off of the coupling, the integral is still negative (green line). Thus, the ANEC is violated and if we send a signal from the right boundary towards the left, it will pass through the horizon of the black hole, it will encounter the negative energy, gain a time advance and finally reappear on the left boundary (see figure 12).

Building traversable wormholes from Casimir energy and non-local couplings

UNIVERSITY OF AMSTERDAM

Master Thesis